Law of Cosines Homework problem

[Turloughmack]

My course: Non- Euclidean Geometry

I have been asked to prove the first law of cosines

cosh(c) = cosh(a).cosh(b) - sinh(a).sinh(b).cos(gamma)

I am told initially to proceed as follows:

Explain why I may assume that T is contained in D (The hyperbolic disk) and also why
V(c) = 0, V(a) = tanh(b/2) and V(b) = e^(i)(gamma)tanh(a/2)

Here V(a), V(b) and V(c) are all vertexes of the hyperbolic triangle with angles alpha, beta and gamma.

I didnt know how to draw the picture here. Anyway could anybody help me on this part of the problem

Thanks!

 

[pat_connell]

Assuming that T is contained in D (The hyperbolic disk) is reasonable since the hyperbolic disk is a space of constant negative curvature, and the First Law of Cosines applies when the space is curved. The coordinates V(c) = 0, V(a) = tanh(b/2) and V(b) = e^(i)(gamma)tanh(a/2) can be easily derived from the definitions of alpha, beta and gamma. Specifically, V(c) is the origin (0,0) since it is the vertex of angle gamma, V(a) is the point tanh(b/2) along the x-axis and V(b) is the point e^(i)(gamma)tanh(a/2) along the circle centered at the origin with a radius equal to tanh(a/2).